The problem of the effect of internal heat evolution on the motion of a heated solid spherical particle in a viscous fluid is analytically solved in the Stokes approximation at small Reynolds and Peclet numbers. The temperature drop between the surface of the particle and the area away from it is assumed to be arbitrary. In solving hydrodynamic equations, the thermal conductivity of the particle is set to be a power function of temperature and the viscosity of the fluid, an exponential-power function of temperature. The observability of this effect is discussedyesBelgorod State Universit

Malay, N. V.

Pleskanev, A. A.

Shchukin, E. R.

DSpace at Belgorod State University

Effect of Internal Heat Evolution on the Motion of a Solid Particle in a Viscous FluidN. V. Malay*, A. A. Pleskanev*, and E. R. Shchukin*** Belgorod State University, Belgorod, 308015 Russia e-mail: malay@bsu.edu.ru ** Institute of High Temperatures Scientific Association (IVTAN), Russian Academy of Sciences, Izhorskaya ul. 13/19, Moscow, 127412 Russia Received June 7, 2005Abstract—The problem of the effect of internal heat evolution on the motion of a heated solid spherical particle in a viscous fluid is analytically solved in the Stokes approximation at small Reynolds and Peclet numbers. The temperature drop between the surface of the particle and the area away from it is assumed to be arbitrary. In solving hydrodynamic equations, the thermal conductivity of the particle is set to be a power function of tem­perature and the viscosity of the fluid, an exponential-power function of temperature. The observability of this effect is discussed.INTRODUCTIONWe consider the steady motion of a solid nonuni- formly heated hydrosol particle of radius R  in a viscous incompressible fluid occupying the whole space. The particle is heated by internal heat sources nonuniformly distributed over its volume with density qir. 0), where r  and 0 are spherical coordinates (0 < 0 < n). Heating of a hydrosol particle may be associated with a chemical reaction proceeding in its volume, its radioactive decay, electromagnetic radiation absorption, etc. As a result, the surface temperature of the particle may far exceed the environmental temperature. This, in turn, may change the electrophysical properties of the fluid and, thereby, substantially affect the velocity and pressure field distributions in the neighborhood of the particle. Of all fluid transport coefficients, the dynamic viscosity is the most temperature-dependent [1]. This depen­dence is given by the formula1 + Z F 4 r £ - 1 exp<j - A [ ^ - l  I ,(1)which describes the variation of the viscosity in a wide temperature range (at Fn = 0, this formula can be reduced to the well-known Reynolds formula [1]). Here, A and Fn are constants, |i„ = \\.e(Tx), and Tx is the fluid temperature away from the particle. For water, we have A = 5.779, = -2.318, F2 = 9.118, and Tx =273 K; for glycerol, A = 17.29, F 1 = -1.228, F2 = 7.022, and Tx = 303 K. The relative error of formula (1) is no higher than 3% in both cases. Factors Fn were calcu­lated using the Maple VIII software package. Hereafter, indices and refer, respectively, to externalparameters (those of the viscous fluid), internal param­eters (those of the particle), and parameters at infinity (away from the particle).Interacting with the nonuniformly heated surface, the fluid starts moving over the surface toward higher temperatures. This effect, called the thermal slip, causes additional force F? to arise. Once this force becomes equal to the viscous force of the environment, the particle starts moving uniformly. Thus, if the tem­perature drop near the particle is significant (i.e., (Tis -  TX)ITX - 0 ( 1 ) ,  where Tis is the mean temperature of the particle surface in the viscous fluid), the solid particle moves steadily.When the particle is heated through electromagnetic radiation absorption, its steady motion in a fluid is termed photophoresis [2, 3]. In this case, the tangential component of the velocity on the surface of a solid aerosol particle (r = R) satisfies the slip condition [4],Un = K t,Ve d r e■r t , de (2)Here, Ue is the shear component of mass velocity U in the spherical coordinate system; v e is the dynamic vis­cosity of the fluid; Te is the fluid temperature; and Kts is the thermal slip coefficient, which is derived from the kinetic theory of gases. If the accommodation coeffi­cients for the tangential momentum, a T. and energy, a i:, equal unity, gaskinetic coefficient = 1.152 [4].MALAY et al.The problem of fluid slip over the surface of a solid hydrosol particle was first solved by Basset [5]. He sup­posed that the tangential velocity of a fluid relative to the surface of a solid is proportional to shear stresses and named the corresponding proportionality coeffi­cient (Kls) the slip coefficient. It is assumed that this coefficient depends on only the nature of the fluid and solid surface (if it is other than zero). If the spherical particle is at rest and the fluid flows about it, this hypothesis for the axisymmetric flow (the Basset hypothesis) takes the formU = K  u ( dU 9 i ldU r  e tsH  dr r 90UerV P e = |teAUe + 2(V(xe • V )U e + [V(xe x curlU J;divUe = 0;A Te = 0; d iv (^ V r ,)  = -qc,(1.1)(1.2)(1.3)Ve dT er = R: Ur = 0, Ue = K ts^ r ^ ,  Te = T h>: U, £/Mcos0er - £ / Msin0ee, (1.4)■P T  -1 co> -*• e —  0 :■T„,Elaboration of a theory accounting for the effect of internal heat evolution on the motion of a solid spheri­cal particle in a viscous nonisothermal fluid is a chal­lenging task. First, the motion of a particle in a fluid is governed by both surface and volume effects arising from nonuniform distributions of hydrodynamic and temperature fields. Second, this motion is a result of slip of the fluid over the solid surface. Thermal slip of a fluid over a solid surface is not yet fully understood. The reason is that a rigorous mathematical theory of inhomogeneous fluids is still lacking. Third, a mecha­nism of thermal energy distribution inside the solid that causes its nonuniform heating is unknown. However, impressive theoretical and experimental advances in this respect have recently been made [6, 7].1. PROBLEM DEFINITIONWe consider the steady motion of a solid nonuni­formly heated spherical hydrosol particle of radius R  in a viscous incompressible fluid occupying the whole space. The particle is heated by heat sources nonuni­formly distributed over its volume with density qt(r, 0). It is assumed that any phase transition is absent, the thermal conductivity of the particle far exceeds the thermal conductivity of the fluid, and the thermal con­ductivity of the particle is a power function of temper­ature (A., = )• Here, = K i T j  and /, = Theparticle moves in the spherical coordinate system (r, 0, (p) with the origin at its center.In view of the above assumptions, the equations and boundary conditions for velocity U,„ pressure Pe, and temperatures Te and T. in the spherical coordinate sys­tem have the form [8, 9],Here, Ur and Ue are the radial and shear component of mass velocity Ue, LL is the velocity of the plane-paral­lel flow of the fluid about the particle (IL  || 0Z), e(. and ee are the unit vectors of the spherical coordinate sys­tem, Ux = |U J, a 0 is the Stefan-Boltzmann constant, a , is the integral blackness, and A is the Laplacian.The basic parameters of the problem are coefficients p,„ |i„. and X,,. as well as quantities R. Tx , and Ux, which remain unchanged during the motion of the spherical particle. From these parameters, one can constitute a dimensionless combination, the Reynolds number, Re„ = (\i„UJ{)/pe <  1. Let us nondimensionalize Eqs. (1.1)—(1.3) and boundary conditions (1.4) as fol­lows: V„ = I ]JU„, tk = 7’/./7’„. and p„ = PJP„ (P,„ = (\x,JJ,JIR), where k = e or i.At 8 = Re„ <s? 1, the incoming flow can be consid­ered as a disturbance; therefore, a solution to the equa­tions of fluid dynamics and heat transfer should be sought in the formVe = v i 0) + 8 v i1)+ ..., pe = p?' + ep™ + ..., , = i ' + V,■(1.5)2. TEMPERATURE FIELDS AND THE DRIVING FORCE AND VELOCITY OF THE STEADY MOTION OF THE HYDROSOL PARTICLESubstituting (1.5) into the set of Eqs. (1.1)—(1.3) and separating the variables, we arrive at expressions for the mass velocity components and temperature fields,Vr(y,Q) = co s0 (l + A 1G1(y) + A 2G2(y)),te(y, 0) = t(e \ y )  + ^ \ y ,  0);Ve(y,Q) = - s in 0 ( l  + A xGz(y) + A 2G4(y)),ti(y,Q) = t\0)(y) + et\1)(y,Q),(2.1)(2.2)wherek(Dy  n=0( «  + 3 )y1 “ a(2)g 2 = - i y ^ ^EFFECT OF INTERNAL HEAT EVOLUTION“ 2>' ,.o(n + 3 ) ln -  -  1y -k(D/ r\ \ 2 n(n + 3) yG3 = G 1 + y-G [, G4 = G 2 + ^G !2,t f \ y ) = l  + l ,  V = ^ tzR \(2.3)t? \y )  =y1 + coBn + -,— "1— I qjdV + J —dy4nRXi„T„yJd V1 + COf(D = _ ^ o s 0 _  (1 + co)^y \ - 2d y - - 2^ f l dy, y = r/R,(1 + (ù)RJBy +3,  = —i;COS0,+ 1. ( , K (1 +CÛ) 2 l n +  1 f ( f\\ D , Wf n(y) =  j - j ;— y — J qi(r,Q)Pn(x)dx,iC° °° -1= ^JîiZrfV'When calculating coefficients A^ ,11 and A^ 2) by these formulas, one should take into account that A*',1 1 = -3 , A<2) = -1 , A f  = 1, C  = a<4) = 1, a<3) = -4 ,od1} = Fn> a i2) = (4 -  n)Fn + AFn_v  a[,2) = 4, a!„4) =A nln !, a^3) = 2AFn _ 1 -  2(n + 2)Fn,A[2) = - J [  6 a (14) + 2 (3 a (11)- a (12)) + a (13)],a  = ^ { - 6 yc44) + [3 (4 a (11) -  a f-1) + a (13)]A(12)- [2 ( 3 a ^ 1)- a ^ 12)) + oc23)]y}.The constants of integration appearing in (2.1) and (2.2) are found from the boundary conditions on the particle surface. Then, integrating the stress tensor over the particle surface, one can find total force F acting on the hydrosol particle, which is a linear combination of viscous force Ffl and force F?,F = F„ + eF , (2.4)whereFn =  - 6 n R \ i „ U „ f ]ln i , F q = 6nR\L ^fqJ n z,U  = ^ exP {_A^ }’f q = K x4 v e G13 / . -V 6 / . ; / '■exp{-Ay},is the dipole moment of the heat source density, Pn(x) are Legendre polynomials, x = cos0, and z = rcos0.In (2.3), G[ and G1' are the first and second deriva­tives of the corresponding functions with respect to y (k = 1, 2). Coefficients A ^  (n > 1) and A^ 2) (n > 3) are found with the recurrence relationsA(1) = -  1k = 1a (2) = - l(n + 3)(n -  2)- 6 a  <ny n)+ ^ i ( n  + 2 - k )N l \y = l = G lG‘2- G 2G\, N 2\y = l = -G[,( 0)1ly = i ’Xn(n + 5)x ^  [(« + 4 -  k ) (a ^ \n  + 5 - k ) -  a f ]) +S  —  1 i n " e ° °  i ® 0 ® 1 ^ 7 i 3  3  t  t  (O0 — 1 + 2^  + 4   ^ Àjs — À;co?es,and n: is the unit vector in the Z direction. Subscript “s” refers to physical quantities taken on the hydrosol par­ticle surface at mean dimensionless temperature /es = T J T X. This temperature is given byT± e;f i= 1 + l4 kR K ~T 'Q)dVGqGiRT«k = 1 KVT  \4 1 er"(2.5)-  1x [(n + 2 -  k ) -  a i2)] + <xf] }y kA ^ lk+ [(2n + 5 -  2fc)ai1) -  a f )]ykA(„1)_k_2k = 0In (2.5), integration is over the entire volume of the particle.With the condition Xe <  satisfied, one can ignore the 0 dependence of the dynamic viscosity in the parti-cle-fluid system and put | i = |i,,( i^'  ). Then, expresMALAY et al.x 10-/„, W/cm2Function /zph vs. incident radiation intensity 70 for Cû = (7) 0.5, (2) 0.7, and (3) 1.0. Curve 4  is constructed for small temperature drops.sion (1) takes the formYi +  £ F'„.1 yexp-j - A (2.6)Formula (2.6) allows splitting the problem into ther­mal and hydrodynamic subproblems. Respective solu­tions are joined together using the boundary conditions on the surface of the hydrosol particle.Equating total force F to zero yields an expression for the velocity of steady motion (steady-state velocity) of a solid spherical particle in a viscous nonisothermal fluid,U, = - e fV n z.J LI(2.7)thickness 5R < R  adjacent to the area being heated and the heat source density within this layer is given by [10]4;(r, 0) = <- Û -0, o < e < ï ,where I0 is the incident radiation intensity.Then,J ?i(r, Q)zdV = k R 3I 0, J ?i(r, Q)dV = k R 2I0,y yand we obtain expressions accounting for the effect of internal heat evolution on the driving force and velocity of steady motion of a solid spherical particle acting as an absolutely black body,Fvh = -E6izR\L„KiafjA1, Uvh = eK tshvh,wheref v h2 v e G13 t esN 1b l mT,v es Gxex p { -A y } /0,^ph  - ■/n.3. RESULTS AND DISCUSSIONFormulas (2.4) and (2.7) make it possible to esti­mate the effect of internal heat evolution on the driving force and velocity of steady motion of a hydrosol parti­cle when the viscosity of the surrounding fluid is an exponential-power function of temperature.It follows from these formulas that the sense and magnitude of F? and U? depend on the sense and mag­nitude of the dipole moment of the heat source density, as well as on the thermal conductivity of the particle. At A., — ► 00  and a fixed value of the dipole moment of the heat source density, F? and U? tend to zero.To estimate the effect of heat sources nonuniformly distributed over the volume of the hydrosol particle on its steady-state velocity, it is necessary to know their nature. Let heat sources arise when the particle acting as a black body absorbs electromagnetic radiation. In this case, the radiation is absorbed in a thin layer ofThe figure plots function /;ph against incident radia­tion intensity I0 to illustrate the effect of internal heat evolution on phoretic velocity Uph of the solid hydrosol particle. Numerical estimates were made for borated graphite particles suspended in water. The calculation parameters were T^ = 273 K, k,,„ = 55 W/(m K), and three values of co. Curve 4 in the figure was constructed for small relative temperature drops (y — ► 0, co = 0) but for molecular transport coefficients taken at the particle surface mean temperature. Internal heat evolution is seen to significantly influence the steady-state velocity of the hydrosol particle.CONCLUSIONSWe derived expressions for the force acting on a spherical hydrosol particle and for its steady-state velocity at an arbitrary temperature drop between the particle surface and the surrounding fluid away from the particle. The particle is heated by heat sources non­uniformly distributed in its volume. The results obtained in this work may be helpful in designing experimental setups imposing directional motion on hydrosol particles, estimating the sedimentation rate of hydrosol particles in channels, analyzing transport of hydrosol particles in the chemical reaction zone, etc. Quantitative investigation into this effect for solid par­ticles seems to be a feasible task.EFFECT OF INTERNAL HEAT EVOLUTIONREFERENCES1. S. Bretshneider, Properties o f Gases and Liquids: Cal­culation Techniques for Engineers (Khimiya, Moscow, 1966) (translated from Polish).2. V. B. Kutukov, E. R. Shchukin, and Yu. I. Yalamov, Zh. Tekh. Fiz. 46, 626 (1976) [Sov. Phys. Tech. Phys. 21, 361 (1976)].3. E. R. Shchukin and N. V. Malai, Inzh.-Fiz. Zh. 54, 628 (1988).4. Yu. I. Yalamov, A. B. Poddoskin, and A. A. Yushkanov, Dokl. Akad. Nauk SSSR 254, 1047 (1980) [Sov. Phys. Dokl. 25, 734 (1980)].5. K. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, Englewood Cliffs, 1965; Mir, Moscow, 1976).6. S.A. Beresnev and L. B. Kochneva, Opt. Atmos. Okeana 16, 134 (2003).7. L. A. Dombrovskii, Teplofiz. Vys. Temp. 42, 961 (2004).8. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow, 1986; Pergamon, New York, 1987).9. G. Z. Gershuni and E. M. Zhukhovitskii, Convective Sta­bility of Incompressible Fluids (Nauka, Moscow, 1972; Springfield, Jerusalem, 1976).10. C. F. Bohren and D. R. Huffman, Absorption and Scat­tering of Light by Small Particles (Wiley, New York, 1983; Mir, Moscow, 1986).Translated by V. IsaakyanCopyright of Technical Physics is the property of Springer Science & Business Media B.V. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.

Effect of internal heat evolution on the motion of a solid particle in a viscous fluid

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Effect of internal heat evolution on the motion of a solid particle in a viscous fluid

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